Lifting the j-invariant: Questions of Mazur and Tate

In this paper we analyze the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We show that its coordinates are rational functions on the j-invariant of the elliptic curve in characteristic p. In particular, we prove that the second coordinate is always regular at j=0 and j=1728, even when those correspond to supersingular values. A proof is given which yields a new proof for some results of Kaneko and Zagier about the modular polynomial.